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There are few questions like here and here already asked about this. But I don't have the background to understand the answers there. I am just beginning to learn classical algebraic geometry, and don't yet have the understanding of even the basic notions like schemes. So I am asking the question again.

In the context of a ramified covering map for Riemann surfaces, the formula relates the Euler characteristics of two surfaces. More precisely if $\pi : X \to Y$ is a complex analytic covering map between two Riemann surfaces, and if the degree of $\pi$ is $N$, then we have

$$ 2-2g_X = N(2-2g_Y) - R $$

where

$g_X, g_Y$ are the genus of $X,Y$ respectively,

$R=\sum_{p\in Y}(e_p-1)$ is a finite sum over the points of $X$ at which $\pi$ is ramified and $e_p$ denotes the ramification index.

Now to my actual questions:

$1)$ What if $X,Y$ are connected complex manifolds of dimension greater than $1$? How does the formula read? (An explanation as elementary as possible will be greatly appreciated.)

$2)$ Why can't the same formula as above be used for manifolds of higher dimension?

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  • The setting in the higher-dimensional case is similar to the case of curves: you start with a reasonable map $f:X\to Y$ (say proper with finite fibres) and you want to relate topological invariants of $X$ to topological invariants of $Y$. Unfortunately, one of the points made in the answers to the questions you link to is that you should forget about "numerical relations between topological invariants" and rather work with equalities of line bundles (or divisors). So maybe try to learn a bit about canonical bundles and understand the meaning of Riemann-Hurwitz: $K_X = f*K_Y + R$. – Ariyan Javanpeykar Mar 26 '15 at 15:17
  • Yes, if you are referring to the adjunction formula, I am familiar with it. –  Mar 26 '15 at 17:37
  • I have an interesting nontrivial answer to your question on fixed points in projective spaces (which generalizes to ${\rm GL}(V)$ acting on $\Bbb P(V)$) that I think would be useful and helpful to have on the site. Did you already figure out the answer to your question? I do not know why it was downvoted. – anon Mar 30 '15 at 01:43

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